R help - Oct 2002 - cubic spline smoothers with heterogeneous variances

Hello. I have data (plant weights over time) that are non-linear and in 
which the variance increases over time.  I have to estimate the first 
derivatives of plant weight given time (i.e. growth rate) and their se, 
using a regression smoother, and I have been considering cubic spline 
smoothers.  However, I do not know if this can be done given that the error 
variance would increase over time.  Does anyone know what the effect of a 
non-constant error variance has on the estimates of the 1st derivative and 
its se?

Bill Shipley
Departement de biologie
Universite de Sherbrooke
Sherbrooke (Quebec) CANADA J1K 2R9
Bill.Shipley at USherbrooke.ca
http://callisto.si.usherb.ca:8080/bshipley/

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>>>>> "Bill" == Bill Shipley <Bill.Shipley at
Usherbrooke.ca>
>>>>>     on Tue, 22 Oct 2002 14:49:34 -0700 writes:
    Bill> Hello. I have data (plant weights over time) that are
    Bill> non-linear and in which the variance increases over
    Bill> time.  I have to estimate the first derivatives of
    Bill> plant weight given time (i.e. growth rate) and their
    Bill> se, using a regression smoother, and I have been
    Bill> considering cubic spline smoothers.  

fine. I do so too if I need derivatives.

    Bill> However, I do not know if this can be done given that
    Bill> the error variance would increase over time.  

I'd hope that a simple two-stage procedure (possibly iterated)
would be enough :

1. Smooth(x,y) with ``df = small'' (depend on your context),
   i.e. getting a smooth solution.
2. Get the residuals and  Smooth(x, abs(resid)) 
   to get an estimate proportional to sigma(x).
3. Smooth(x, y,  weights = 1 / sigma(x))

{now you could iterate "2." and "3." and hopefully see
 convergence (of some kind)}.

    Bill> Does anyone know what the effect of a non-constant error
    Bill> variance has on the estimates of the 1st derivative
    Bill> and its se?

"adverse" (effects), but hopefully you'd only look at the 1st
derivative after the above 2-stage solution.

Martin Maechler <maechler at stat.math.ethz.ch>
http://stat.ethz.ch/~maechler/
Seminar fuer Statistik, ETH-Zentrum  LEO C16	Leonhardstr. 27
ETH (Federal Inst. Technology)	8092 Zurich	SWITZERLAND
phone: x-41-1-632-3408		fax: ...-1228			<><
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Sorry for coming to this so late.  My understanding of the problem seems to
be different from Martin's.  Hopefully someone can set me straight.

In Martin's Step 1, why use a small df?  I thought if the objective is to
estimate the variance function first, then one would want estimate of the
regression function to have as small a bias as possible (so that the
residuals would consist of mostly error and very little bias).  I believe
that was the motivation behind those difference-based estimators of the
residual variance.

Also, I was under the impression that the estimation of derivatives has been
worked on quite a bit using local polynomials, so why not use those?  One
just need a variable bandwith smoother for heteroscedastic error.

Cheers,
Andy

-----Original Message-----
From: Martin Maechler [mailto:maechler at
stat.math.ethz.ch]
>>>>> "Bill" == Bill Shipley <Bill.Shipley at
Usherbrooke.ca>
>>>>>     on Tue, 22 Oct 2002 14:49:34 -0700 writes:
    Bill> Hello. I have data (plant weights over time) that are
    Bill> non-linear and in which the variance increases over
    Bill> time.  I have to estimate the first derivatives of
    Bill> plant weight given time (i.e. growth rate) and their
    Bill> se, using a regression smoother, and I have been
    Bill> considering cubic spline smoothers.  

fine. I do so too if I need derivatives.

    Bill> However, I do not know if this can be done given that
    Bill> the error variance would increase over time.  

I'd hope that a simple two-stage procedure (possibly iterated)
would be enough :

1. Smooth(x,y) with ``df = small'' (depend on your context),
   i.e. getting a smooth solution.
2. Get the residuals and  Smooth(x, abs(resid)) 
   to get an estimate proportional to sigma(x).
3. Smooth(x, y,  weights = 1 / sigma(x))

{now you could iterate "2." and "3." and hopefully see
 convergence (of some kind)}.

    Bill> Does anyone know what the effect of a non-constant error
    Bill> variance has on the estimates of the 1st derivative
    Bill> and its se?

"adverse" (effects), but hopefully you'd only look at the 1st
derivative after the above 2-stage solution.

Martin Maechler <maechler at stat.math.ethz.ch>
http://stat.ethz.ch/~maechler/
Seminar fuer Statistik, ETH-Zentrum  LEO C16	Leonhardstr. 27
ETH (Federal Inst. Technology)	8092 Zurich	SWITZERLAND
phone: x-41-1-632-3408		fax: ...-1228			<><
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